# Bernoulli Equation and gauge pressure

• Zahid Iftikhar
In summary, the question is asking for the gauge pressure required in the city mains for a stream from a fire hose connected to the mains to reach a vertical building of height 15m. The Bernoulli Equation can be used to solve this problem, assuming two points along the path of the water flow. The solution depends on whether the water is delivered through a large supply pipeline or inside a hose. In the first case, the velocity in the main pipe is essentially zero and the height of the water fountain will be determined by the pressure difference between the main pipe and atmospheric pressure. In the second case, the water flow velocity is constant along the entire length of the hose, but the water level inside the hose depends on the pressure in the
Zahid Iftikhar

## Homework Statement

What gauge pressure is required in the city mains for a stream from a fire hose connected to the city mains to reach a vertical building of height 15m?

## Homework Equations

Bernoulli Equation

## The Attempt at a Solution

I have tried this sum. My confusion is regarding choice of points where Bernoulli equation is to be applied. I assume two points,one at the bottom with height h1, P1 and v1, and the other at top of the building with h2=15m, P2=0 (as water is supposed to go this much height only) and v2=0 (again water is supposed to be stationary as this is the maximum height). I can't understand how is v1=v2, this is the condition that provides the required answer. Some of my work is given in the attached image please. I shall be grateful for the reply.

#### Attachments

• my work.jpeg
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Delta2
Maybe you should assume that the firemen are taking the hose to the top of the roof?

#### Attachments

• FireHose_01.jpg
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The velocity in the main is essentially zero, because it is a big huge pipe. What does this give you?

stockzahn said:
Maybe you should assume that the firemen are taking the hose to the top of the roof?
Thanks for the reply. Let's assume there is large sized supply pipeline that has an outlet of any cross section where pressure is required to be found as is shown by fig you added. We attach a vertical hose 15m tall to reach the top of the building,again as per fig. If the Velocities at top and bottom of the are to be same, how will pressure at bottom be minimum. In this case water may go further up. To me at top of building both pressure and velocity have to be zero. Now if we apply Bernoulli eq. then data provided is not sufficient. Pl comment.

Assuming the case #1 (you've drawn):

As Chestermiller mentioned you could assume that the main supply pipe's diameter is sufficiently large to neglect the flow velocity. Then the velocity of the initial point in the main pipe is zero and the height of the generated water fountain corresponds to the pressure difference between the main pipe and the atmospheric pressure. This of course only works, if the water jet "spreads" to fulfil the mass conservation - the jet cross section must increase to diminish the velocity.

Assuming case #2:

If the water is delivered inside a hose to the roof, then it cannot spread, therefore the flow velocity must be constant along the entire length. The flow velocity though depends on the hydraulic head (the water level in the hose), hence the height of the building.

If the pressure in the main duct corresponds exactly to the "height" of the building, in case #1 the water fountain will reach the edge of the roof. In case #2 the hose will be filled entirely with water, but none of it will exit the hose and its level will be a the nozzle. In both cases the initial velocity and the final velocity are identical zero.

Delta2
stockzahn said:
Assuming the case #1 (you've drawn):

As Chestermiller mentioned you could assume that the main supply pipe's diameter is sufficiently large to neglect the flow velocity. Then the velocity of the initial point in the main pipe is zero and the height of the generated water fountain corresponds to the pressure difference between the main pipe and the atmospheric pressure. This of course only works, if the water jet "spreads" to fulfil the mass conservation - the jet cross section must increase to diminish the velocity.

Assuming case #2:

If the water is delivered inside a hose to the roof, then it cannot spread, therefore the flow velocity must be constant along the entire length. The flow velocity though depends on the hydraulic head (the water level in the hose), hence the height of the building.

If the pressure in the main duct corresponds exactly to the "height" of the building, in case #1 the water fountain will reach the edge of the roof. In case #2 the hose will be filled entirely with water, but none of it will exit the hose and its level will be a the nozzle. In both cases the initial velocity and the final velocity are identical zero.
Thanks a lot for the time. I need to ponder over the points you have reflected.

Delta2

## 1. What is the Bernoulli Equation?

The Bernoulli Equation is a fundamental principle in fluid dynamics that describes the relationship between the velocity, pressure, and elevation of a fluid in a system. It states that in a steady flow, the sum of kinetic energy, potential energy, and pressure energy remains constant.

## 2. How is the Bernoulli Equation derived?

The Bernoulli Equation is derived from the principles of conservation of energy and mass. It utilizes the assumptions of steady flow, incompressibility, and negligible friction to simplify the equations and arrive at the final form of the equation.

## 3. What is gauge pressure?

Gauge pressure is the pressure measured relative to atmospheric pressure. It is the difference between the absolute pressure and the local atmospheric pressure. It is commonly used in engineering and fluid dynamics to measure pressure in closed systems.

## 4. How is gauge pressure related to the Bernoulli Equation?

The Bernoulli Equation includes the term for static pressure, which is the pressure exerted by a fluid at rest. Gauge pressure is used to measure this static pressure in a closed system. By incorporating this measurement into the Bernoulli Equation, it allows for the calculation of various fluid properties such as flow rate and velocity.

## 5. What are some real-world applications of the Bernoulli Equation and gauge pressure?

The Bernoulli Equation and gauge pressure have numerous applications in various fields such as aviation, hydraulics, and fluid mechanics. Some examples include calculating the lift force on an airplane wing, predicting flow rates in piping systems, and designing hydraulic systems for construction equipment.

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